Nowhere-zero 3- ows in squares of graphs

Nowhere-Zero 3-Flows in Squares of Graphs

It was conjectured by Tutte that every 4-edge-connected graph admits a nowherezero 3-flow. In this paper, we give a complete characterization of graphs whose squares admit nowhere-zero 3-flows and thus confirm Tutte’s 3-flow conjecture for the family of squares of graphs.

Nowhere-zero ows in random graphs

Tutte's 5-flow conjecture for graphs of nonorientable genus 5

We develop four constructions for nowhere-zero 5-ows of 3-regular graphs which satisfy special structural conditions. Using these constructions we show a minimal counterexample to Tutte's 5-ow conjecture is of order 44 and therefore every bridgeless graph of nonorientable genus 5 has a nowhere-zero 5-ow. One of the structural properties is formulated in terms of the structure of the multigraph ...

Nowhere-zero 3-flows in triangularly connected graphs

Let H1 and H2 be two subgraphs of a graph G. We say that G is the 2-sum of H1 and H2, denoted by H1 ⊕2 H2, if E(H1)∪E(H2)=E(G), |V (H1)∩ V (H2)| = 2, and |E(H1)∩E(H2)| = 1. A triangle-path in a graph G is a sequence of distinct triangles T1T2 · · ·Tm in G such that for 1 i m − 1, |E(Ti) ∩ E(Ti+1)| = 1 and E(Ti) ∩ E(Tj ) = ∅ if j > i + 1. A connected graph G is triangularly connected if for any ...

Inside-Out Polytopes

We present a common generalization of counting lattice points in rational polytopes and the enumeration of proper graph colorings, nowhere-zero flows on graphs, magic squares and graphs, antimagic squares and graphs, compositions of an integer whose parts are partially distinct, and generalized latin squares. Our method is to generalize Ehrhart’s theory of lattice-point counting to a convex pol...